Compact weak G₂-manifolds with conical singularities

We construct 7-dimensional compact Einstein spaces with conical singularities that preserve 1/8 of the supersymmetries of M-theory. Mathematically they have weak G_2-holonomy. We show that for every non-compact G_2-holonomy manifold which is asymptotic to a cone on a 6-manifold Y, there is a corresponding weak G_2-manifold with two conical singularities which, close to the singularities, looks like a cone on Y. Our construction provides explicit metrics on these weak G_2-manifolds. We completely determine the cohomology of these manifolds in terms of the cohomology of Y.